Over the past three decades, an increasing amount of literature has emerged on the topic of sufficient dimension reduction (SDR). [1]} proposed the sliced inverse regression to reduce the dimension of the predictors. That is, assuming the response only depends on a linear combination of the predictors, one regresses the predictors \(\mathbf {X}\)  against the response \(\mathbf {Y}\)  to circumvent any model-fitting process. [2]} defined the central subspace as the subspace with the minimal dimension
such that the response is independent of the predictors given the projection of the predictors onto the space. Other SDR methods include but not limited to sliced average variance estimation [3]}, principal Hessian direction [4]}, contour regression [5]}, inverse regression estimation [6]}, directional regression [7]}, likelihood-acquired directions [8]}, discretization-expectation estimation [9]}, non-elliptically distributed predictors [10]}, [11]}, dimension reduction based on canonical correlation [12]}, [13]}, and average partial mean estimation [14]}. However, the aforementioned methods all focus on the dimension reduction of predictors with univariate response.
